All social beings need to communicate and use language, even if it's a very basic language.
Dogs communicate through many kinds of language, for instance. They bark, move, run, etc to show if they're happy, hungry, sad, etc. That is a form of language for itself in my opinion.
Therefore, I can say that language came first; culture actually depends on language and sharing information.
We express our culture through language. Even the simplest gesture can be considered language.
Formally language can be "any system of formalized symbols, signs, sounds gestures, or like used of conceived as a means of communicating thought, emotion, etc." but it can also be "the means of communication used by animals".
So, as proved by this definition, language came first than culture.
Irrational animals also have a system of sounds and gestures used to communicate thought, emotion, etc, so they officially use and always used language to hunt, mate, escape from predators, etc.; by other words, to survive.
If your definition of language is only verbal and written language that is developed through culture and its study is by itself a culture, then you can say culture came first. That isn't the total and complete definition of language though.
Human language is the most complex in the world but dolphins also have a very complex language system. Through sound waves they can communicate every feeling they have to others.
"Culture can be defined as all the ways of life including arts, beliefs and institutions of a population that are passed down from generation to generation. () , it includes codes of manners, dress, language, religion, rituals, norms of behavior such as law and morality, and systems of belief as well as the art." From Wikipedia, the free encyclopedia
That seems like a very human definition.
So, culture is only possible between humans. That is a good argument to show that language came first because language is possible in the animal kingdom and culture is not.
We began as irrational beings so we had no culture, but we had language because we were able to communicate.
In conclusion, language is older than culture because language is fundamental for survival has social beings and culture is not. Animals live without culture and that's why they don't have evolution. All generations live the same way and do the same: they live, mate, and die.
So, language is needed for survival, culture is needed for evolution.